pair functions applied to Zeta function in two ways

*To*: mathgroup at smc.vnet.net*Subject*: [mg52513] pair functions applied to Zeta function in two ways*From*: Roger Bagula <tftn at earthlink.net>*Date*: Tue, 30 Nov 2004 05:24:31 -0500 (EST)*Reply-to*: tftn at earthlink.net*Sender*: owner-wri-mathgroup at wolfram.com

When I thought of this , I felt that it might work for the zeta zeros, but experience with it in Mathematica shows these sums functions are too far from analytical in s=1/2 strip. (* pair functions applied to Zeta function in two ways*) (*the first with {1/(n+1),n/(1+n)} as the pair gives nearly a straight line as parametricPlot*) (* second with {1/(x+1),x/(1+x)} as the pair gives seems like conic curve:parabolic or hyperbolic*) (* The second has a definite minimum due to the gzetax near 3.5*) (* it is doubtful these sums will be any good in the zeta zero region of 1/2+I*b[n]*) (* where fzeta[1/2+I*b[n]]=-gzeta[1/2+I*b[n]]: fzetax[1/2+I*b[n]]=-gzetax[1/2+I*b[n]] *) fzeta[x_]=Sum[If[Mod[n,2]==1,(1/(n+1)),(n/(n+1))]/n^x,{n,1,Infinity}] gzeta[x_]=Sum[If[Mod[n,2]==1,(n/(n+1)),(1/(n+1))]/n^x,{n,1,Infinity}] ParametricPlot[{N[fzeta[x]],N[gzeta[x]]},{x,2,5}] Plot[N[fzeta[x]],{x,2,5},PlotRange->All] Plot[N[gzeta[x]],{x,2,5},PlotRange->All] fzetax[x_]=Sum[If[Mod[n,2]==1,(1/(x+1)),(x/(x+1))]/n^x,{n,1,Infinity}] gzetax[x_]=Sum[If[Mod[n,2]==1,(x/(x+1)),(1/(x+1))]/n^x,{n,1,Infinity}] ParametricPlot[{N[fzetax[x]],N[gzetax[x]]},{x,2,5},PlotRange->All] Plot[N[fzetax[x]],{x,2,5},PlotRange->All] Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn at netscape.net URL : http://home.earthlink.net/~tftn